On the asymptotic behaviour of the Vasconcelos invariant for graded modules
Luca Fiorindo, Dipankar Ghosh
TL;DR
The paper extends the Vasconcelos invariant (the v-number) from homogeneous ideals to finitely generated graded $R$-modules by interpreting $v(M)$ as the initial degree of certain annihilator quotients, and proves that, for a homogeneous ideal $I$ and submodule $N$ of a graded module $M$, the v-numbers and indegrees of powers and quotients such as $I^nM$, $I^nM/I^{n+1}M$, and $M/I^nN$ become linear in $n$ for large $n$. The leading coefficients are explicitly described in terms of the degrees $d_1\le\cdots\le d_c$ of a reduction $J=(y_1,\dots,y_c)$ of $I$, with the critical index $\delta$ determined by the annihilator structure in a corresponding bigraded Rees-module setting. The results generalize and strengthen prior work by Conca (domain case) and Ficarra-Sgroi (polynomial case), and they provide precise conditions under which the asymptotic $v$-values of module quotients align, including explicit equalities $v_\mathfrak p(I^nM/I^{n+1}N)=v_\mathfrak p(M/I^{n+1}N)$ in large $n$ when certain stability holds. The techniques hinge on Rees modules and bigraded (or multigraded) structures, enabling a unified, explicit description of leading terms and stability of associated primes.
Abstract
The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behaviour of the minimum distance of projective Reed-Muller type codes. We initiate the study of this invariant for graded modules. Let $R$ be a Noetherian $\mathbb{N}$-graded ring, and $M$ be a finitely generated graded $R$-module. The v-number $v(M)$ can be defined as the least possible degree of a homogeneous element $x$ of $M$ for which $(0:_Rx)$ is a prime ideal of $R$. For a homogeneous ideal $I$ of $R$, we mainly prove that $v(I^nM)$ and $v(I^nM/I^{n+1}M)$ are eventually linear functions of $n$. In addition, if $(0:_M I)=0$, then $v(M/I^{n}M)$ is also eventually linear with the same leading coefficient as that of $v(I^nM/I^{n+1}M)$. These leading coefficients are described explicitly. The result on the linearity of $v(M/I^{n}M)$ considerably strengthens a recent result of Conca which was shown when $R$ is a domain and $M=R$, and Ficarra-Sgroi where the polynomial case is treated.